3 Mixtures of Simple Liquids By I. R. McDONALD Department of Chemistry Royal Hollowa y College Englefield Green Surrey 1 Introduction Some very remarkable advances have been made in the theory of simple liquid mixtures in the comparatively short period since the topic was last reviewed in the pages of Annual Reports.’-3 Much of this progress may be ascribed to three factors first a variety of theoretical approaches which had previously been applied successfully to the problem of pure fluids including in particular, theories of the perturbation type have now been extended to include the case of mixtures ; secondly the n-fluid theories which grew out of the important concept of the random mixture have been replaced by improved models based in part on the ideas of van der Waals ; and thirdly the results of large-scale Monte Carlo computations for mixtures of Lennard-Jones molecules have become available.The subject matter of the present review is limited to a discussion of the calculation of excess thermodynamic properties of binary mixtures of molecules which interact via spherically symmetric pair-wise additive potentials and for which quantum effects are negligible. Specific reference is made to two potential models. The first of these is the hard-sphere potential defined by the relations 1 q,j(r) = coy Y < d, = 0 Y 2 dij where dij is the hard-sphere diameter for species i andj. Quite cleslrly this poten-tial does not adequately represent the interaction between real molecules. In particular because there is no attractive region to the potential the hard-sphere fluid can have no true liquid phase and the only phase transition is that between solid and fluid.A second more realistic model is the Lennard-Jones 12-6 potential : where cij is the depth of the potential well and oij is the collision diameter i.e. qi,(oij) = 0. A mixture of 12-6 fluids provides an example of a conformal mixture, ‘ N. G. Parsonage Ann. Reports ( A ) 1967,64 57. N. G. Parsonage Ann. Reports ( A ) 1968 65 33. I. R. McDonald and K. Singer Ann. Reports ( A ) 1970 67 45. 7 76 I. R. McDonald i.e. one in which all pair potentials have the form q ( r ) = ~F(r/a) where E and a are parameters and the function F is the same for all interactions. The goal of a theory of mixtures is not so much the calculation of the properties of the mixture itself but the properties of mixing.In the case of liquid mixtures the properties of greatest interest are the excess thermodynamic functions for mixing at constant pressure the excess Gibbs free energy GE excess enthalpy HE and excess volume VE. It is with the calculation of these quantities that the theories reviewed here are largely concerned. Recent experimental results are discussed in Section 2 and theoretical work is described in Sections 2-5 ; com-parison between theory and experiment is largely reserved for Section 6. The material is similar to that in a recent Specialist Periodical Report4 but more emphasis is placed on the latest developments. Other accounts of the theory of mixtures may be found in two recent publications uiz.the second edition of the monograph by Rowlinson’ and an authoritative review article by Henderson and Leonard. 2 Experimental Results It is convenient to use the word experimental to refer both to measurements made on real liquid mixtures and to computer ‘experiments’ on model systems7 In practice it is found that computer simulation provides the more useful data against which to test theories. The reason for this is that the machine calculations are free of any uncertainty about the form of the various pair potentials in the mixture. The simulation work is of course the only source of experimental data on systems of hard spheres. Extensive studies in which computer experiments have played an important role,* have shown that the Lennard-Jones 1 2 4 potential is a satisfactory eflectiue pair potential for simple liquids.The latter group includes not only the heavier noble-gas liquids (Ar Kr and Xe) but also liquids composed either of small diatomic molecules e.g. N2 0, and CO or of quasi-spherical molecules such as CH and CF,. On the other hand it is now known that the true pair potential in such systems is certainly not of the 12-6 form. In the case of Ar for example, the pair potential between isolated atoms has a deeper and narrower bowl and a weaker tail than the 12-6 potential which best reproduces the bulk properties of the liquid.9-” This apparent anomaly may be resolved by including the effects of weak and predominantly repulsive three-body forces. However it is also fair to say that the thermodynamic properties of the liquid are not very I.R. McDonald in ‘Statistical Mechanics’ ed. K. Singer (Specialist Periodical Reports), The Chemical Society London 1973 vol. 1 . J. S. Rowlinson ‘Liquids and Liquid Mixtures’ Butterworth London 2nd edn. 1969. D. Henderson and P. J. Leonard in ‘Physical Chemistry An Advanced Treatise’ ed. H. Eyring D. Henderson and W. Jost Academic Press New York 1971 vol. VIII, ch. 7. ’ I. R. McDonald and K. Singer Quart. Rev. 1970 24 38. * I. R. McDonald and K. Singer Mol. Phys. 1972 23 29. J. A. Barker and A. Pompe Austral. J . Chem. 1968 21 1683. l o M. V. Bobetic and J. A. Barker Phys. Rev. (B) 1970 2 4169. * J. A. Barker R. A. Fisher and R. 0. Watts Mol. Phys. 1971 21 657 Mixtures of Simple Liquids 77 sensitive to the precise form of the pair potential and that a number of different potentials may be made to fit the experimental data by choosing suitable values for the interaction parameters.Despite its known inadequacies the 12-6 potential is a common choice in theoretical work on liquids primarily because of its simple analytical form. In applying the 12-6 potential to mixtures the interaction parameters for species 1 and 2 are most frequently determined from the well-known combining rules : = ( E ~ ~ E ) * (Berthelot rule) (3) cr12 = +(oll + G,,) (Lorentzrule) (4) Such a system is called a Lorentz-Berthelot mixture. The parameter c12 may, in principle be determined directly from measurements of either the cross second virial coefficientI2 or the composition dependence of the critical temperature.l 3 The measurements which exist are not very precise but the general trend suggests strongly that E ' is very nearly always some one to two per cent less than that given by the geometric mean of E~~ and E,,. For this reason a number of other combining rules have been proposed,14*15 but it is often more useful to retain the general form of equation (3) and write The quantity k is a pure number of the order of unity which may be treated as an adjustable parameter and varied so as to force agreement between theory and experiment for one selected property of the mixture. In the past less doubt has been cast on the validity of the arithmetic-mean rule for cr1 equation (4) which is exact in the case of hard spheres. Recently it has been suggested by Good and Hope'67 l 7 that better results on cross virial coeffi-cients may be obtained if a geometric-mean rule is used.The experimental data which they use to support their argument all refer to mixtures of relatively complex molecules. It is possible of course to treat both E and o12 as variable quantities and thereby to obtain agreement between theory and experiment for a wider range of properties but is questionable whether any great significance can be attached to the results of such manipulations. Henderson and Leonard6 have given an excellent summary of the experimental measurements of excess thermodynamic properties of mixtures of real simple liquids which covers the period up to 1969. To this should be added the measure-ments made some years ago on N2 + CH4.'8i'9 Systems of interest for which I ' M.A. Byrne M. R. Jones and L. A. K. Staveley Trans. Faraday SOC. 1968,64 1747. '' 1. W. Jones and J. S. Rowlinson Trans. Faraday SOC. 1963,59 1702. l 4 G. H. Hudson and J. C. McCoubrey Trans. Faraday SOC. 1960 56 761. I s B. E. F. Fender and G. D. Halsey J . Chem. Phys. 1962,36 1881. '' R. J. Good and C. J. Hope J . Chem. Phys 1970,53,540. " R. J. Good and C. J. Hope J . Chem. Phys. 1971,55 1 1 1. l 9 S. Fuks and A. Bellemans Bull. Soc. chim. belges 1967 76 290. F. B. Sprow and J. M. Prausnitz Amer. Ins?. Chem. Engineers J. 1966 12 780 78 I. R. McDonald data have since been reported include Kr + Xe,20*2' Ar + Kr,21 Kr + CH4,22 and Ar + CH4.23 The situation at present is that data are available on GE HE, and VE (not necessarily all at the same temperature) for Ar + Kr Ar + N2, Ar + CH, O2 + Ar O2 + N2 CO + CH4 and Kr + CH4 and on GE and VE alone for Kr + Xe Ar + CO N2 + CO N2 + CH, and CH4 + CF,.The values reported by different authors are often in poor agreement with each other, particularly when account is taken of measurements made over a range of temperature and it is clear that some of the published results are significantly in error. Added to the inaccuracies in the measurement of excess properties are the inevitable uncertainties about the detailed form of the pair potentials. This combination of factors means that it is rarely possible to make unambiguous comparisons between experimental data on real systems and the predictions of different theoretical approaches.In particular it is known that small departures from the Lorentz-Berthelot rules may result in large changes in the magnitude, and even a change in sign of the excess properties. Much greater reliance may be placed on data obtained from computer experiments based either on the method of molecular dynamics or on the Monte Carlo method. Machine calculations of this type provide what is essentially exact information on the consequences of a given intermolecular force law. Their success arises primarily from the fact that a model containing a relatively small number of particles (usually several hundred with suitable boundary conditions) is in general found to be sufficient to simulate the behaviour of a macroscopic system. As the model may be chosen to correspond to any set of parameter values for a given potential it is possible to study systematically the relation between interaction parameters and the thermodynamic properties of the mixture and thereby to determine the range of applicability of different theories.In the method of molecular dynamics the classical equations of motion of a system of interacting particles are solved numerically and equilibrium properties are determined from time averages taken over a sufficiently long time interval (- 10- s). The Monte Carlo procedure involves the generation of a series of configurations of the particles of the model in a way which ensures that the configurations are distributed in phase space according to some prescribed probability density. The mean value of any configurational property determined from a sufficiently large number (- lo6) of configurations provides an estimate of the ensemble-average value of that quantity.The character of this ensemble average depends upon the chosen probability density. Most applications of the Monte Carlo method to the study of fluids have been based on the usual Gibbs petit-canonical constant volume or NVT-ensemble but work has also been carried out based on the isothermal-isobaric or NpT-ensemble. The NpT-ensemble is a natural choice for the study of liquid mixtures particularly of the J. C. G. Calado and L. A. K. Staveley Trans. Faraday SOC. 1971,67 289. 2 1 C. Chui and F. B. Canfield Trans. Faraday SOC. 1971 67 2933. 2 2 J. C. G. Calado and L. A. K. Staveley Trans. Faraday SOC. 1971 67 1261 23 J.C. G. Calado and L. A. K. Staveley J . Chem. Phys. 1972 56,4718 Mixtures of Simple Liquids 79 excess properties because the results required are almost invariably the changes which occur on mixing at constant (usually near-zero) pressure. Data obtained by the NVT-method may be processed in such a way as to provide information on changes in thermodynamic properties in the constant-pressure mixing process but this requires additional rather tedious calculations. The NpT-method has its own disadvantages but it does have the merit of yielding the required results in an attractively direct manner. In practice however the two methods are found to yield data of approximately equal statistical reliability for a given expenditure on computing time and the choice of which to use is largely a matter of taste.Results for mixtures of hard spheres were obtained some years ago both by molecular dynamics24 and by the Monte Carlo meth~d.~’-~’ The major advan-tage of molecular dynamics is that it allows the study of time-dependent pheno-mena. For the calculation of equilibrium properties the Monte Carlo method is generally more suitable and published results on excess properties for the 12-6 potential model have all been obtained in this Two lengthy series of Monte Carlo computations have recently been made for binary mixtures of 12-6 liquids one in the NpT-en~emble~~* 31 and the other based on the NVT-meth~d.~’ In most of the calculations the parameters gl and nl2 (given by the Lorentz-Berthelot rules) were held constant and the ratios E ~ J E and 611/422 were systematically varied.Further reference to these results will be made in Section 6 where the reliability of different theories is discussed. Results have also been obtained for some model systems simulating real mixtures and these are shown in Table 1. The agreement between the NpT- and NVT-calculations is good. Agreement between computation and experiment on the other hand is generally rather poor. Some improvement may be achieved by varying the quantity k,, and the values of k which are listed in Table 2 are those required to bring the calculated values of GE into agreement with experiment. Though some dis-crepancies remain the results for HE and VE based on the modified values of k12 are overall in substantially better agreement with experiment.It can be seen from Table 2 that the adjusted value of k, is less than unity for all mixtures except Ar + N2 and the small positive deviation found for this system is in-significant in comparison with the combined errors in the experiments and the computations. Thus the conclusion to be drawn both from liquid-state measure-ments and from the analysis of gas-phase data is that it is the rule rather than the exception for to be less than that given by the geometric-mean relation, equation (3). 2 4 B. J. Alder J . Chem. Phys. 1964 2724. ’’ 2 6 E. B. Smith and K. R. Lea Trans. Furaday SOC. 1963,59 1535. 2 7 A. Rotenberg J . Chem. Phys. 1965,43,4377. ” I. R. McDonald Chem. Phys. Letters 1969 3 241. 2 9 1. R. McDonald Mol. Phys.1972 23 41. 3 0 J. V. L. Singer and K. Singer Mol. Phys. 1972,24 357. 3 1 E. B. Smith and K. R. Lea Nature 1960 186 714. I. R. McDonald Mol. Phys. 1972 24 391 00 0 Table 1 Comparison4 between experimental excess thermodynamic properties of equimolar mixtures and the predictions of various theories p = 0 k, = 1 System T/K experiment (a) GE/J mol-MC-NpT" MC-NVT~ RM' APMd vdW 1 vdW2f PYP Pert.h Var.' (b) HE/J mol-experiment MC- Np T MC-NVT RM APM vdWl vdW2 PY Pert. Var. Ar + Kr 116 + 84 +46+7 + 45 + 203 + 139 + 46 + 61 +41 + 39 + 47 Ar + N Ar + 0, 84 84 + 34 + 37 +35+5 - + 35 0 + 98 + 57 + 39 + 27 + 32 + 42 ----- ---- +51 + 60 -29+17 +16*9 -- 18 +40 0 + 254 + 142 -+ 162 + 78 - 30 + 43 + 18 + 28 - 30 - 28 + 35 - 31 + 42 ---- ---Ar + CO 84 + 57 +26+5 + 27 + 99 + 55 + 29 + 20 + 25 + 28 --+37? 15 + 34 + 149 + 80 + 35 + 22 + 30 + 37 -Ar + CH, 91 + 74 -14&6 - 10 + 227 + 159 - 17 +9 - 12 - 12 -Xe + Kr N +O CO +CH CH,+CF, 161 84 91 111 +.115 + 39 +115 + 360 +381t5 +77&7 - + 28 + 38 + 76 - 28 - + 143 + 137 -- + 80 +111 - + 43 + 83 + 29 + 84 + 36 + 69 -- -- -- - - -- -- + 76 - -- + 103 -60$12 -- 35 - 52 +406 - + 223 -- 55 -11 -- 30 - 34 -- ---+ 42 +39& 15 + 48 + 220 +118 + 52 + 33 + 42 -+ 105 -+15+12 - + 13 + 59 + 96 + 26 + 52 - - *+ + 24 + 30 - + 115 --- 4 - 2 -0 3 (c) VE/cm3 rnol-' experiment MC-NpT MC-N VT RM APM vdWl vdW2 PY Pert.Var. -0.52 - 0 69 j- 0.06 - u.60 + 0.42 + 0.09 - 0.68 - 0.47 - 0.62 - 0.73 --0.18 +0.14 -0.25k0.05 -- 0.23 0.00 +0.11 -- 0.02 --0.25 -- 0.20 - 0.23 - 0.26 -- ---+ 0.10 +0.17 -- 0.1 7 & 0.05 - 0.22 & 0.04 --0.17 -0.13 - 0.62 + 0.23 + 1.34 - + 0.05 + 0.6 1 --0.19 - 0.23 --0.16 -C.17 --0.17 -0.12 --0.17 -0.14 -- - -;i g F 2 % ki - 5 tcl 6' E - & -0.31 - 0.32 + 0.88 - 0.28 & 0.06 - 0.76 & 0.06 -+0.35 -0.32 -+ 0.38 - 0.30 - 0.28 -0.71 -- 0.24 -0.50 -- 0.25 - 0.63 - 0.25 - 0.68 -0.10 $ - --- -0.75 -Monte Carlo-NpT method. * Monte Carlc -NVT method. Random-mixing approximation. ' Average Potential Model. One-fluid van der Waals model.Two-fluid van der Waals model. Percus-Yevick theory. ' Perturbation method. Variational method Table 2 Comparison between experimental excess thermodynamic properties of equimolar mixtures and the results of Monte Carlo calculations,p = 0 k # 1 System Ar + Kr Ar + N Ar + 0, T/K 116 84 84 (a) k , MC-NpT 0.989 1.001 -MC-NVT 0.989 1 .Ooo 0.988 (b) HE/J mol-* experiment - +51 + 60 MC-NpT +29 + 34 -MC-NVT +36 + 40 + 52 (c) vE/cm3 mol-' experiment -0.52 -0.18 +0.14 MC-N V T - 0.5 3 - 0.23 + 0.06 - MC-NpT - 0.60 - 0.25 Values of k are adjusted to fit experimental data on GE. Ar + CO 84 0.989 0.989 - + 79 + 79 +0.10 -0.10 -0.11 Ar + CH Xe + Kr 91 161 0.975 -0.976 0.98 1 - + 103 + 57 +83 --+0.17 - 0.70 -0.11 -- 0.01 - 0.47 N2 + 0, 84 0.999 1 .Ooo - + 42 + 42 -0.31 - 0.28 - 0.25 CO + CH CH + CF, 91 111 0.988 -0.988 0.900 + 105 -- + 70 - + 67 - 0.32 - 0.88 - 0.68 --0.61 + 0.8 Mixtures of Simple Liquids 83 3 Distribution Function Theories If the intermolecular potentials are spherically symmetric a knowledge of the pair distribution functions gijr) is sufficient to determine the equation of state of the mixture.However the calculation of the functions gijr) is a formidable task. In the case of pure liquids the most widely used theory of this type is that due to Percus and Y e ~ i c k . ~ ~ The extension of their approach to the problem of mixtures is straightforward and has been considered by a number of author^.^^-^^ It is convenient first to define the total correlation function hijr) as hiAr) = gijr) - 1 (6) The total correlation function is related to the direct correlation function ci,{r) through a generalization of the well-known Ornstein-Zernike relation : where p = N / V is the number density and the sum is taken over all components (labelled k) in the mixture.The Percus-Yevick approximation is given by ci,{r> = gijr)(1 - ex~[qi,{r)/kTI) (8) Substitution of (8) into (7) yields a set of coupled non-linear integral equations for the functions g&). When these are solved the thermodynamic properties of the mixture may be calculated uia one of several different routes the well-known statistical mechanical expressions for the pressure (the virial theorem) the com-pressibility and the configurational energy may all be used.These expressions all have the form of integrals with respect to r of functions involving g&) and if the g&) were exact they would yield identical results. In practice slightly different results are obtained because the approximation described by equation (8) has been introduced. L e b ~ w i t z ~ ~ and B a ~ t e r ~ ~ have solved the Percus-Yevick equations exactly for hard-sphere mixtures with additive diameters i.e. systems for which dl2 = 241 + d22) (9) and have succeeded in obtaining the equation of state in closed form. Comparison with the simulation results of various workers shows that p” (the solution of the compressibility equation) and p’ (the solution of the virial or pressure equation) bound the true pressure at all densities in the fluid range.35 An excellent fit to the machine calculations is obtained36 by taking a linear combination of the 32 J.K. Percus and G. J. Yevick Phys. Rev. 1958 110 1. 3 3 J. L. Lebowitz. Phys. Rev. 1964 133 A895. 3 4 R. J. Baxter J. Chem. Phys. 1970 52 4559. 3 5 J. L. Lebowitz and J. S . Rowlinson J. Chem. Phys. 1964,41 133. 36 G. A. Mansoori N. F. Carnahan K. E. Starling and T. W. Leland J. Chem. Phys., 197 1,54 1523 84 I. R. McDonald solutions : p = $pc -k $pv This represents a generalization to the case of mixtures of a similar proposal made for the one-component hard-sphere fluid by Carnahan and Starling.37 Mansoori et al.36 have used equation (10) to evaluate the excess thermodynamic properties of a binary mixture in which x1 = x2 = 3 and dl1/dzz = 3 for comparison with the molecular dynamics data of Alder.24 Agreement is found to be good even at this large value of the diameter ratio and the authors conclude that equation (10) represents the best available analytical equation of state for hard-sphere mixtures.The results themselves are of interest for the light they may throw on the behaviour of real systems. There is a pressure drop on mixing at constant volume which increases rapidly with increasing density. Mixing at constant pressure leads to a contraction in volume at all ratios of molecular diameters and all densities in the fluid range. It follows that GE is also invariably negative and consequently there is no phase separation of the components. The absolute value of GE is small which suggests that in real mixtures large values of GE must be associated with differences in the attractive forces.Equation (10) could also be used in calculations for real systems by incorporating it into a generalized equation of state of the van der Waals type. 38-42 The Percus-Yevick equations cannot be solved analytically for the 1 2 4 potential and numerical methods must be used. The pair distribution functions and excess thermodynamic properties of a number of 12-6 mixtures at low den-sities and high temperatures were calculated some years ago by Throop and Bearman.43* 44 The systems were chosen to represent binary mixtures formed from Ne Ar and Kr but qualitatively the results were the same in all cases. For mixing at constant pressure GE VE and the excess internal energy are all positive at low densities increase to a maximum with increasing density and then decrease ; both GE and V E become negative at high densities.The excess functions for mixing at constant volume are much smaller than for the constant-pressure process. Only recently have Percus-Yevick calculations been attempted in the liquid range. Grundke Henderson and Murphy45* 46 have obtained results for a system representing an equimolar mixture of Ar and Kr at the triple-point temperature of Kr (116 K). The thermodynamic functions were obtained by integrating the equation for the configurational energy. This is known to be the most accurate route to the thermodynamic properties in the case of pure and thus presumably for mixtures also.3 7 N. F. Carnahan and K. E. Starling J . Chem. Phys. 1969 51 635. 3 8 M. Rigby Quart. Rev. 1970 24 416. 39 H. C. Longuet-Higgins and B. Widom Mol. Phys. 1964 8 549. 40 M. L. McGlashan Trans. Faraday SOC. 1970,66 18. 4 1 K. N. Marsh M. L. McGlashan and C. Warr Trans. Faraday SOC. 1970,66,2453. 4 2 N. S. Snider and T. M. Herrington i. Chem. Phys. 1967 47 2248. 4 3 G. J. Throop and R. J. Bearman J . Chem. Phys. 1966,44 1423. 44 G. J. Throop and R. J. Bearman J . Chem. Phys. 1967 47 3036. 4 5 E. W. Grundke D. Henderson and R. D. Murphy Canad. J . Phys. 1971,49 1593. 4 6 E. W. Grundke D. Henderson and R. D. Murphy Canad. J . Phys. in the press. 4 7 J. A. Barker D. Henderson and R. 0. Watts Phys. Letters ( A ) 1970 31 48 Mixtures of Simple Liquids 85 49 has been extended by Gibbons5’* 5 1 to include mixtures of hard convex particles of arbitrary shape.Scaled-particle theory is an example of a distribution function theory because the quantity evaluated which determines the equation of state is the value of the pair distribution function at contact. The procedure used is to write down an expression for the reversible work needed to create a cavity within the fluid and relate this to the average density of particles at the boundary of the cavity. The equation of state which is derived in this way involves the volume the surface area, and the mean radius averaged over all orientations of particles of each species in the mixture. When the values appropriate to hard spheres are substituted the result reduces to that obtained from the Percus-Yevick theory via the com-pressibility equation.Results have been obtained for mixtures of hard spheres, cylinders cubes tetrahedra and ellipsoids. It is found that VE is always negative and so therefore is GE. The effect of shape on VE is much less than that of size ratio. The apparent lack of any simple relationship between the shape and VE at a fixed temperature and pressure suggests that there is no equivalent sphere whch can reproduce the mixing properties of a non-spherical particle. Little effort has yet been made to interpret the properties of mixtures of real molecules in the light of these results. The scaled-particle theory of hard-sphere 4 n-Fluid Theories Until comparatively recently most statistical thermodynamic treatments of liquid mixtures were based on what have become known as n-Juid models.In theories of this type the properties of a binary mixture are taken to be those of either a single imaginary pure fluid (a one-fluid model) or of an ideal mixture of two imaginary components (a two-fluid model); a three-fluid model has also been proposed but has not been widely used for liquid mixture^.^^^'^ The inter-molecular potentials which characterize the hypothetical substances are formed by taking some suitable composition-dependent average of the potentials in the various components of the real system and theories of t h s class differ from each other in the way in which the average potentials are computed. Practical calculations are simplified if the pair potentials in the mixture and the average potentials are conformal with each other because the thermodynamic properties of mixing may then be obtained from those ofa single reference system by applica-tion of the law of corresponding states.For this reason most calculations for real systems have been based on the 1 2 4 potential model. The various n-fluid models which have been proposed have been put forward on the basis of a variety of arguments. Henderson and Leonard6. 5 3 have been 4 8 H. L. Frisch Adv. Chem. Phys. 1964 6 229. 4 9 D. Henderson and S. G. Davison in ‘Physical Chemistry An Advanced Treatise’ ed. H. Eyring D. Henderson and W. Jost Academic Press New York 1967 vol. 11, chap. 7. R. M. Gibbons Mof. Phys. 1970 18 809. ’’ R. M. Gibbons Mof. Phys. 1969 17 81. 5 1 5 2 D.Henderson and P. J. Leonard Proc. Nu,. Acad. Sci. U.S.A. 1970,67 1818. 5 3 D. Henderson and P. J . Leonard Proc. Nut. Acad. Sci. U.S.A. 1971,68 632 86 I. R. McDonald able to show however that the essential features of all the widely used models can be expressed very simply in terms of molecular distribution functions. For example the assumption expressed by equation (1 I ) generates the (one-fluid) random-mixing approximation of Prigogine and others.5L62 The quantity gx(r) is interpreted as the pair distribution function in a hypothetical pure fluid called the equivalent substance which is characterized by a composition-dependent pair potential cp,(r) = &,F(r/aX) given by i j The rather more plausible assumption, leads to the two-fluid Average Potential Model of which a detailed account has been given by Bellemans Mathot and Simon.63 The quantity gll(r) is now interpreted not as the pair distribution function for species i in the mixture but as that in a pure fluid having a pair potential The two fluids in the model are those characterized by the pair potentials q J r ) (i = 1 or 2) with parameters The random-mixing rule equation (12) may also be deduced by assuming that all permutations of molecules amongst positions in a given configuration are equally probable.This is a poor approximation when the molecules differ in size because it ignores the size-ordering effect which is the most important factor determining the structure of such systems. The inadequacy of the model is particularly obvious in the case of hard-sphere mixtures because in general the interchange of a large sphere with a small one leads to molecular overlaps and hence to states of infinite potential energy.Even when the potential does not have a hard core the effect of the approximation is a large and spurious contribu-tion to the free energy in mixtures of different size. In fact M a n ~ o o r i ~ ~ and others60i65 have been able to show that for conformal mixtures the random-and axi. 5 4 I. Prigogine A. Bellemans and A. Englert-Chowles J . Chem. Phys. 1956 24 518. 5 5 I. Prigogine 'The Molecular Theory of Solutions' North-Holland Amsterdam 1957. " W. Byers Brown Phil. Trans. Roy. SOC. 1957 A250 175. '' W. Byers Brown Phil. Trans. Roy. SOC. 1957 A250 221. s g W. Byers Brown Proc. Roy. SOC. 1957 A240 561.5 9 Z. W. Salsburg and J. G. Kirkwood J . Chem. Phys. 1952 20 1538. '' R. L. Scott J . Chem. Phys. 1956 25 193. '' Z. W. Salsburg P. J. Wojtowicz and J. G. Kirkwood J . Chem. Phys. 1957 26 1533. '' P. J. Wojtowicz Z . W. Salsburg and J. G . Kirkwood J . Chem. Phys. 1957 27 505. " A. Bellemans V. Mathot and M. Simon Adv. Chem. Phys. 1967,11 117. 6 4 G. A. Mansoori J . Chem. Phys. 1972 57 198. ' 5 R. M. Mazo J . Chem. Phys. 1964,40 1454 Mixtures of Simple Liquids 87 mixing approximation provides an upper bound on the free energy of the real system. The defects inherent in the assumption of random mixing are only partially overcome by the introduction of the approximation (13) and use of the Average Potential Model also results in an unrealistically large positive contribu-tion to the free energy of mixing except in cases when the molecules are nearly equal in size.Neither the random-mixing approximation nor the Average Poten-tial Model can be usefully applied to hard-sphere mixtures. A more accurate description of mixtures is obtained by supposing that the g&) are functions which are related to each other on a corresponding-states basis. Specifically it is assumed that (1 5 ) sll(r/%) = S12(1/612) = g22(dc22) = g,(r/g,) say It is clear that this approximation takes into account the ordering effects of size differences. In the case of conformal mixtures equation (15) can be shown to lead to a prescription for &,a in a one-fluid theory. In order to define the equi-valent substance uniquely a further expression is required.By applying equation (15) to the case of hard spheres at contact an expression may be derived for d,, the diameter of the hard spheres in the equivalent hard-sphere system in terms of the d i j . It is then assumed that the same functional relationship holds between a and aij. This approximation is called the one-fluid van der Waals model because the mixing rules for E and ox are a modem version of van der Waals’ pro-posal for the calculation of the parameters in his equation of state for mix-tures.66-68 A two-fluid van der Waals follows if it is assumed by analogy with equation (13) that g12(r/012) = 3[Sll(Y/alJ + gz2(r/a22)1 (16) The whole question of one-fluid theories has recently been discussed by Srnith6’v7’ from the point of view of the conformal solution theory of Longuet-Higgins and Byers B r ~ w n .~ ” ~ ~ The basis of conformal solution theory is the expansion of the Helmholtz free energy of the mixture about that of a pure reference fluid which is taken to be conformal with the components of the mix-ture in powers of some suitably chosen expansion variable. Let the pair potentials in the mixture and in the reference fluid be characterized respectively by para-meters qj aij and E’ no. In order to maintain maximum generality Smith69 6 6 T. W. Leland P. S. Chappelear and B. W. Gamson Amer. Inst. Chem. Engineers J . , 6 7 6 a T. W. Leland J. S. Rowlinson and G. A. Sather Trans. Faraday SOC. 1968,64 1477. 69 W. R. Smith Canad. J . Chem. Eng. 1972 50 271. ’O W. R. Smith Mol. Phys. 1971 22 105.’’ H. C. Longuet-Higgins Mol. Phys. 1958 1 83. 72 W. Byers Brown and H. C. Longuet-Higgins Proc. Roy. SOC. 1951 A200,416. 1962 8 482. R. C. Reid and T. W. Leland Amer. Inst. Chem. Engineers J. 1965 11 228 88 I. R. McDonald chooses to make the expansion in powers of a = ~ ’ d and b = ~ ‘ c . 9 and finds that A = A + NkTCxiInxi 1 + RL2) c xixJaij - a,) + R r ) c xix,(bij - b,) + higher-order terms i j i j where A is the free energy of the reference fluid. The coeficients Ri2) and Ri2) involve integrals over the pair distribution function of the reference system ; the higher-order terms become rapidly more complex and even the second-order terms involve integrals over the three- and four-body distribution functions. In order to make use of this result it is necessary to choose both a reference fluid and a set of values for p q r and s.The first-order term in equation (17) was originally derived by L~nguet-Higgins~’ who chosep = 1 s = 3 q = r = 0. This choice makes it possible to express the coefficients Ri2) and R f ) in terms of thermodynamic functions of the reference system which Longuet-Higgins took as being one of the pure components. The same choice of the reference fluid but with p = s = 1 4 = r = 0 has been made more recently by Tan and L ~ k s . ~ ~ These authors succeeded in evaluating the second-order terms in equation (17) by re-writing the most complicated coefficients in terms of the density and tem-perature derivatives of the pair distribution function. However the use of one pure component as the reference system or of any reference system which is independent of concentration means that the perturbation cannot be small at all composition^.^^ A better choice of reference system is one which annuls the first-order terms in equation (17).This requires that E and 0 be chosen such that Conformal solution theory therefore leads naturally to a one-fluid theory of the corresponding-states type. In first order the properties of the mixture (apart from the ideal free energy of mixing) are those of an equivalent pure substance with a pair potential characterized by the parameters .c0 and 0,. Different choices for p 4 r and s correspond to different mixing rules. For example, p = r = 1 q = 6 s = 12 is the random-mixing approximation for 12-6 potentials and p = 1 r = 0 4 = s = 3 is the van der Waals model.The higher-order terms in equation (17) are corrections to the particular one-fluid theory which is adopted. 7 3 P. Y. Tan and K. D. Luks J . Chem. Phys. 1970 52 3091. 7 4 D. Henderson and P. J. Leonard Proc. Nut. Acad. Sci. U S A . 1971,68 2354 Mixtures of Simple Liquids 89 5 Perturbation and Variational Theories A notable feature of recent theoretical work on liquids has been the development of a number of very successful perturbation methods for the calculation of equili-brium properties. Two of these due to Barker and Henderson and to Mansoori and Canfield have also been used to calculate the excess thermodynamic pro-perties of mixtures again with considerable success. 76 the Helmholtz free energy is expanded about that of a system of hard spheres.There are two principal reasons for choosing hard spheres as the reference system in a perturba-tion treatment. First it is known that the structure of real simple liquid is similar to that of a hard-sphere fluid because the structure is largely determined by the repulsive part of the pair potential. Secondly the equation of state and pair distribution function for the hard-sphere system are available either from machine calculations or from approximate analytical theories. The extension of the theory to the problem of mixtures77y78 may be made in two ways correspond-ing to two different choices for the reference system. Either a single-component hard-sphere fluid or a mixture of hard spheres may be used. The former is required for the treatment of hard-sphere mixtures and yields results in good agreement with computer experiment^^^ but proves unsatisfactory when used for mixtures of 12-6 molecules; only the latter choice will be discussed here.Let cpiJr) be the true pair potentials in the mixture. The starting point of the Barker-Henderson method is the introduction of a set of modified potential functions t,biJr) which are related to the functions cpiJr) in the following way : exp [ - t,biJr)/kT] = 1 - H(d + xij - Rij) exp [ - q i l d i j + xij)/kT] In the Barker-Henderson theory of one-component + H(d, + xij - Rij) + H(r - dij) ( ~ X P [-yijqij(r)/kTI - 11 (19) where xij = (Y - dij)/aij d is the hard-sphere diameter clij varies the steepness of the potential in the repulsive region yij varies the depth of the potential in the attractive region and H ( x ) is the Heaviside step function defined by H(x) = 0 x < 0 = 1 x > o In the case of 12-6 potentials it is convenient to set Rij = aij though this choice is not essential.When aij = yij = 0 the potential t,bijr) becomes the potential for hard spheres of diameter d i j ; when ctij = y i j = 1 $ij(r) is identical to cpiJr). 7 5 J. A. Barker and D. Henderson J. Chem. Phys. 1967,47 2856. 7 6 J. A. Barker and D. Henderson J. Chem. Phys. 1967,47 4714. 7 7 D. Henderson and J. A. Barker J. Chem. Phys. 1968 49 3377. P. J. Leonard D. Henderson and J. A. Barker Trans. Faraday SOC. 1970 66 2439 90 I. R. McDonald The Helmholtz free energy of the mixture of interest may now be expanded in powers of aij and yij.The first-order result is ( A - A,)/NkT = 271p C xixi[ - aij - d$gijo(dij){dij - 6,) i j where the subscript zero is used to denote properties of the reference system and 6 = { 1 - exp [ - qijz)/kT]) dz j:" As the parameters d and d, are arbitrary it is convenient to choose them in such a way as to annul the terms in a and a, on the right-hand side of equation (21) i.e. d = a, d, = a,. Setting aij = y i j = 1 for all i andj the final result becomes ( A - A,)/NkT = -4np c xix,d;gijo(dij)[dij - 6,] i > j + 271(P/k~) 1 xix,Sm qijr)gijo(r)r dr (23) i j R i l In applications to binary mixtures of this version of the theory the Percus-Yevick results for gij,(r) have been used to evaluate A from equation (23). As these results were obtained under the usual assumption of additivity of diameters it follows that d, cannot be chosen to annul the a,,-term.A method which is probably more accurate is to use a non-additive hard-sphere mixture [i.e. a system for which d # $ ( d + d,,)] as a reference fluid. Unfortunately there are no Percus-Yevick or simulation results available for such mixtures so this approach cannot be exploited at present. In the first published series of calculations for mixtures of 12-6 liquids Leonard, Henderson and Barker78 used for A an expression obtained by adding the excess properties calculated by the Percus-Yevick theory to the properties of an ideal mixture of hard spheres. The properties of the two components in the ideal mixture were obtained from the Pad6 approximant to the hard-sphere equation of state which has been proposed by Ree and Hoover.79 However it is now known that use of this procedure gives rise to appreciable errors in the calculated excess properties and more accurate results are obtained" when A, is derived from the equation of state of Mansoori et u I .~ ~ equation (10) of this review. The variational theory of Manssori and Leland". 82 is an extension to mixtures of the work on pure fluids by Mansoori and G~nfield.~~ The starting point is 7 9 F. H. Ree and W. G. Hoover J . Chem. Phys. 1964,40,939. E. W. Grundke D. Henderson J. A. Barker and P. J. Leonard Mol. Phys. in the press. G. A. Mansoori and T. W. Leland J . Chem. Phys. 1970,53 1931. G. A. Mansoori J . Chem. Phys. 1972,56 5335.83 G. A. Mansoori and F. B. Canfield J . Chem. Phys. 1969,51,4958 Mixtures of Simple Liquids 91 an inequality which relates the Helmholtz free energy of a system of interest, which may be referred to as the real system to that of a reference system. This inequality has the form The subscript zero is again used to denote properties of the reference system and the subscripted angular brackets denote an expectation value for states of the reference system. In the case when the real system is a binary mixture the reference system is taken as a mixture having the same composition. If the reference system is characterized by pair potentials qijo(r) and the mixture of interest by pair potentials qi,(r) the inequality (24) may be written in terms of gijo(r) the pair distribution functions in the reference system in the form A < A + 2nNp 1 xixj (25) i j where In all applications of the variational method to mixtures of 12-6 liquids which have been reported up till now the reference system has been taken to be a mixture of additive hard spheres with diameters d and d, .An approximation to the free energy of the real system is then obtained by minimizing the right-hand side of (25) with respect to simultaneous variations in d and d, . This minimization is most conveniently achieved by reformulating the inequality in terms of the Laplace transforms of the hard-sphere pair distribution functions for which analytical relations have been derived33 from the solution of the Percus-Yevick equation. For a specified set of conditions (temperature density intermolecular potential parameters and mole fractions) in the ranges so far studied only one relative minimum is found for the right-hand side of (25).The optimum values of d and d, for a given mixture vary only very slowly with composition. The Helmholtz free energy of the reference system is again most reliably determined from the semi-empirical equation of state equation (10). 6 Comparison Between Theory and Experiment Values of the excess properties of a number of real mixtures which are predicted by the various theories discussed in this review are shown in Table 1. Comparison with the Monte Carlo results which are also listed in Table 1 reveals the in-adequacy of both the random-mixing approximation and the Average Potential Model.On the other hand there is a high correlation between the signs of the excess properties of the real systems and the predictions of these two theories. This fortuitous agreement emphasizes the fact that comparison between theory and laboratory experiments on real mixtures is at best of limited value and may indeed be misleading. In the present case for example a measure of agreement is obtained because errors arising from a spurious positive contribution to th 92 I. R. McDonald free energy of mixing of molecules of differing sizes are partially offset by the neglect of departures from the geometric-mean (Berthelot) rule. The other theories are very much more satisfactory. They are all in agreement with the Monte Carlo results in predicting that VE is invariably negative but that GE and HE may have either sign.From the results given in Table 1 it is difficult, however to draw any firm conclusions regarding the relative accuracy of the different theories and more interest is attached to a systematic study over a range of intermolecular parameter ratios. The results of such a survey are displayed in Figures 1-9. Examination of these graphs leaves no doubt that perturbation theory and variational theory are both reEarkably successful in accounting for the excess properties of mixtures of simple liquids. Not surprisingly bearing in mind their basic similarity the two approaches yield results in broad agreement with each other; variational theory appears to yield slightly more accurate values for GE and HE but perturbation theory is more successful in the prediction of VE.However it is also true that the good results for the excess properties are to some extent the consequence of a canceIlation of large errors Figures 10 and 11 show that neither theory is able to predict accurately the total properties of either the pure components or the mixture. Figures 1-9 show also that the van der Waals one-fluid model is clearly superior to the more sophisticated two-fluid version. The one-fluid theory works well over a moderately wide range of parameter ratios but under extreme conditions large errors occur; this is particularly noticeable in the case of VE. The parameter range for which the theory gives good results is sufficiently wide to include all the real systems listed in Table 1 with the exception of CH + CF,.200 0 I I I I 0 9 1 0 1 1 a1 1 012 -Figure 1 GE as a function of 01,/o12 for an equimolar mixture of 12-6 liquids ur 97 K and zero pressure (EIZ/k = 133.5 K t s 1 2 = 3.596 A k = 1.0).'* The points show the results of Monte Carlo calculations and the curvesgive the predictions of various theories: Barker-Henderson perturbation theory (solid curve) variational theory (broken curve), and the van der Waals one-Jluid model (dot-dash curve) The curves are labelled with the appropriate value of E~ 1 / ~ Mixtures of Simple Liquids 300 I w t 200 100 0 93 0 -200 I I I I I I 0.9 1.0 1.1 ‘1 1 QI 2 -Figure 2a as a function of crl ]/a for an equimolar mixture of 12-6 liquids at 97 K and zero pressure.For other details see caption to Figure 1. (The variational results are omitted for the sake of clarity) 0.9 10 1.1 a1 1 -‘17 Figure 2b See caption to Figure 2 94 I. R. McDonald I I I 0.9 1 .o 1.1 Figure 3 V E as a function of a1 l/a12 for an equimolar mixture of and zero pressure. For other details see caption to Figure 1 400 300 w u 100 0 I I I I 0.92 0.96 1.0 1.01 1.08 1 2-6 liquids at 97 K Figure 4 GE as a function of c1 ,/al for an equimolar mixture of 12-6 liquids at 115.8 K and zero pressure (El2/k = 141.4K a12 = 3.59& k, = 1.0).*' The points and the curves have the same meaning as in Figure 1. The curves are labelled with the appropriate value of E ,/ Mixtures of Simple Liquids 95 1=115.8 K -100 0 92 0 96 10 1.04 1.08 Figure 5 HE as a function of c1 Jc12 for an equimolar mixture of 12-6 liquidr at 115.8 K and zero pressure.For other details see caption to Figure 4 O t 1.00 I m - 3 -I 0.92 0.96 10 1 OL 1.08 01 I 612 -Figure 6 VE as a f i c t i o n of o1 Jal2 for an equimolar mixture of 12-6 liquids at 1 15.8 K and zero pressure. For other details see caption to Figure 96 I . R. McDonald 300 200 W u 0 -100 0 9 10 1 011 012 -Figure 7 GE as a function of 0 ,lo for an equimolar mixture of 12-6 liquids at 1 11 K and zero pressure (E12!k = 133.5 K c12 = 3.596 A k, = LO).*' The double dot-dash curve shows the predictions of the van der Waals two$uid model and the points and other curves have the same meaning as in Figure 1. The curves are labdled with the appropriate value of 1 / ~ 1 2 4 00 200 .W a 0 -200 0 9 1 0 1.1 011 012 -Figure8a as a function of ( ~ ~ ~ / u ~ ~ for an equimolar mixture of 12-6 liquids at 117 K and zero pressure. For other details see caption to Figure Mixtures of Simple Liquids 200 1 Figure 8b See caption I r=117 'I -100 O 1. / I /' I I 1 I 0.9 1.0 1.1 u11 012 -to Figure 8a 0 - 1 I - 0 E - 3 97 -L I I I I I I 0.9 1.0 1.1 011 a12 -Figure 9 VE as a function of c1 JcI2 for an equimolar mixture of 12-6 liquids at 117 K and zero pressure. For other details see caption to Figure 7 A number of problems and possibilities remain. It would be desirable for example to place the van der Waals one-fluid model on a more satisfactory theoretical basis and to understand the reasons for its superiority to the two-fluid version. This should make it possible to improve the model in a natural way. Unfortunately there are limits to the applicability of any one-fluid theory 98 I. R. McDonald a2 0.L a6 0.8 1.0 1, Figure 10 Enthalpy as a function of composition.for the system Ar + Kr ut 116 K zero press~re.~' The triangles show the results of Monte Carlo calculations and curves give the predictions of the Barker-Henderson perturbation theory (pert) the variational theory (oar) and the and 361 33t A A A Mixtures of Simple Liquids 99 The reason for this may be understood by considering the equivalent substance in such a model as the reference system in a conformal solution theory. At some stage when the potential parameters for the pure components are sufficiently different it becomes necessary to include second-order terms in the free-energy expansion. Even if it is possible to evaluate these terms the theory has at this point lost the simplicity which is its most attractive feature. The incorporation of higher-order terms in a perturbation or variational treatment may prove to be a more profitable line of advance because the application of these methods is not limited to conformal mixtures